Optimization of a Photographic Shutter by Minimization of Closing Time Variability

Theses

Thesis/Dissertation Collections

1984

Optimization of a Photographic Shutter by

Minimization of Closing Time Variability

Steven D. Daniels

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please [email protected].

Recommended Citation

Approved by:

CLOSING

TIM~

VARIABILITY

by

Steven D. Daniels

A Thesis Submitted

in

partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

in

MECHANICAL ENGINEERING

Pro f •

Ray C. Johnson

---:(-==T:-:-"h

e

s i

'3

.1\d v i

so r)

Prof.

-Prof.

Prof.

-~---,-(Department Head)

D~PARTM~NT

OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CLOSING TIME VARIABILITY

[

______ ---_________________________________ hereby (grant,

jeny) permission to the Wallace Memorial Library, of R.I.T., to

~eproduce my thesis in whole or in part. Any reproduction will

mechanically

driven,

pivoted _opening blade and a _spring

driven,

pivoted _closing blade which is released

by

a magnetic

holding

coil through an armature lever. The magnitude of the

viscous

drag

of air on the _closing blade is approximated, and its effect on the travel time

(lag

time) of the blade is

determined to be

insignificant.

A set of equations is derived _representing the dynamic performance of the

combination of the _closing blade and the armature lever.

These equations are used to

develop

a mathematical

representation of the variation in

lag

time due to

manufacturing tolerances of the parts involved. Then an optimization computer program is devised and utilized which uses an "exhaustive" search technique to locate the minimum

for the variation in

lag

time as a function of five

independent variables. A _previously written "gradient based" optimization computer program is also employed to minimize variation in

lag

time. The exhaustive program yields a reduction in variation of

lag

time of 6 percent while the

gradient based program yields a reduction of 2 percent.

The author would like to express his deepest gratitude

to his wife,

Nancy,

and his _children, Nicole and Tyler for

their support and the sacrifices

they

made in order to

help

complete this work. Without their participation in this

endeavor, this project would not have been completed on time.

A debt is also owed to John 7. Breen for the competetive

stimulus he provided. He was instrumental in the arrangement

of the word _processing equipment _used, and was generous

enough to provide a much needed quiet place to work when the

going got tough.

Thanks also go out to

Ray

C. Johnson for his guidance

and support as advisor on this project. He helped in the

definition of a manageable thesis objective and provided a

valuable _sounding board for ideas. He also was instrumental

in

helping

diagnose last minute problems with comouter

programs.

We are all products of what we aspire to

be,

what we

are able to

be,

and what other people

help

us to be.

LIST OF FIGURES vi

LIST OF SYMBOLS vii

1. INTRODUCTION 1

2. DESCRIPTION OF SYSTEM 4

3. ANALYTICAL APPROACH TO DYNAMICS

3.1 Calculation of Skin Friction

on

Closing

Blade Due to Air 11

3.2 Equation

Relating

the Angular Displacement

of

Closing

Blade and Armature Lever 17

3.3 Determination of Disengagement Point 21

3.4 Approximation for

Coupling

of Armature Lever

and

Closing

Blade 25

3.5 Equation of Motion for First Part of Travel.... 23

3.6 Equation of Motion for Second Part of Travel... 33

3.7 Calculation of Moments of Inertia ^4

3.8 Example Calculation of

Closing

Time 38

4. APPROACHES TO OPTIMIZATION

4.1 The Optimization Problem 42

4.2 Description of Exhaustive Search Method 46

4.3 Description of Gradient Based Search Method.... 51

5. APPLICATION OF OPTIMIZATION PROGRAMS 55

6. RESULTS 58

7. CONCLUSIONS AND RECOMMENDATIONS 59

BIBLIOGRAPHY 62

APPENDIX A: A Program "for

Calculating

the Angular

Displacement of the Armature Lever

Assembly

6**

APPENDIX B: A Program for

Calculating

the Approximate

Polar Moment of Inertia of an Irregular

Planar Shape 64

APPENDIX C: A Program for the Minimization of the

Variability

of a Photographic Shutter

Through the Use of an Exhaustive Search

Technique 65

APPENDIX D: Modser Algorithm Program, P519RE 85

APPENDIX E: Special

Programming

for Use with Modser

Algorithm Program 1^4

APPENDIX F: Output from Exhaustive Search Program. Ill

APPENDIX G: Output from Gradient Based Search

Program 112

2.1 Shutter System 5

3.1 Approximation Used for Skin Friction Calculation.... 12

3.2

Drag

of Smooth Flat Plate

-Skin Friction 13

3.3

Geometry

of Shutter System 19

3.4 Comparison of Calculated vs. Graphical Values of

Disengagement Angle of

Closing

Bla^e 25

3.5 Angular Displacement of Armature Lever vs. Angular

Displacement of

Closing

Blade for Various Cam Angles ">J

3.5 Method for

Calculating

Polar Moment of Inertia of

Irregular Shapes 36

4.1 Illustration of Scan Se?rch Technique 48

D = _Total

skin friction

Cf

=

Drag

coefficient

S =

Wetted area of plate

p

=

Density

V =

Velocity

Re = Reynolds'

Number

L = _Length

^

= _Dynamic

viscosity

Wav = _{Average angular}

velocity

A = _Displacement

angle

t = _Time

Wmx = _{Maximum angular}

velocity

Vmx = _{Maximum linear}

velocity

R = _Radius

V =

Velocity

T = _Torque

Tmx = _{Maximum torque}

0

= Angular displacement of the armature lever _assembly

^

= _{Angular displacement of the}

closing biaH

y = ^iifj

of the cam edge of the _closing blade

xo = _{X dimension} of the contact point between the _closing

blade and the armature lever

yo = _Y dimension of the contact point betwen the _closing

blade and the armature lever

Xo = _X dimension _of the

(xo,

yo) contact point as the

closing blade rotates

X dimension of the _closing blade pivot

Y dimension of the _closing blade pivot

Angle between the vertical and the line _connecting

(xp,

yp) with (xo, yo)

Radius between

(xp,

_yp) and

(xo,

_yo)

Slope of a straight line

Y axis intercept of a straight line

X dimension of the contact point between the _closing

blade and the armature lever

Y dimension of the contact point between the _closing

blade and the armature lever

Angle of rotation of the armature lever at which it disengages from the _closing blade

Angle of rotation of the _closing blade at which it disengages from the armature lever

Mass

Displacement

Damping

coefficient

Torsional _spring constant

Moment of inertia

Angular displacement of the _closing blade

Natural

frequency

Angular displacement of the _closing blade and shutter spring as a function of time

Angular displacement of the shutter _spring when it is

being

held back

by

the armature lever

Angular displacement of the shutter _spring when the closing blade disengages from the armature lever

IA = _Moment of inertia of gear A xp

yp

P

r

M

B

xf

yf

<P

d

t

d

m

X

b

k

I

6

U)

d(t)

6(0)

6(d)

pdB = _{Pitch diameter}

of gear B

&A

= _{Angular displacement}

of gear A.

Ieq

= _Equivalent

moment of inertia

Icb = _Moment

of inertia of the _closing blade

Ial = _{Moment of inertia of the armature lever}

fl(t)=

_{Angular velocity of the}

closing blade as a function of

time

i7(d)

= _Angular

velocity of the _closing blade when it

disengages from the armature lever

0(h)= _{Angular displacement} of the shutter _spring when the

closing blade is covering half of the aperture

In the design of a photographic _system, one of the most important considerations is the _shutter, the device

by

which

control is exercised on the amount of light _energy which reaches the film. When an exposure is made, the shutter is

required to _{open, allowing} light to reach the

film,

and then

close at the proper time to achieve the desired exposure.

The shutter must close in such a _way that it prevents _any

light from _reaching the

film,

for even a small amount of

light can cause an exposure over

long

periods of time while

the camera is not in use.

The shutter has taken on _many forms throughout the

history

of photography. Some of the earliest cameras

required the operator to remove a lens cover to begin the

exposure and to replace it when he judged that the exposure

was completed. In contrast,

today

military pilots are

equipped with electronic shutters built into their helmet

visors which protect them from

being

blinded

by

the flash

from a nuclear explosion. These shutters are made from

lanthanum-modified lead zircoate titanate crystals, and can

go from translucent to opaque in 85 microseconds. The

familiar amateur cameras available to the consumer

today

use

varying

curtains. Impact shutters consist of a single pivoted

shutter blade which covers a small aperture near the lens.

The blade is driven to an open position

by

the impact of

another _part, and it is returned to the closed position

by

a

spring. This type is _usually found in inexpensive cameras

and has a fixed shutter speed.

The cameras that use the impact type of shutter are

still able to produce acceptable photographs because the film

they

use is able to record an image at a wide range of light

levels. With the introduction in recent years of new films

which require a much more precise exposure, it has become

necessary to produce inexpensive cameras which control

shutter speeds accurately. One of the ways this has been

accomplished is with a shutter design of the type considered

in this paper. It consists of two shutter blades, called an

"opening

blade" and a

"closing blade",

which rotate about the

same pivot and are held in contact edge-to-edge

by

a torsion

spring.

Normally

the opening blade covers a small aperture

near the camera lens. When an exposure is _made, an

electromagnetic

"holding

coil"

is energized, which holds back

the _closing

blade,

while the _opening blade is _mechanically

driven to an open position and trapped there. At the same

time,

an electronic sensor is

integrating

the light from the

The exposure times for this type of system are often

shorter than the

lag

time of the _closing blade.

Lag

time is

defined as the time required for the _closing blade to move

from rest to the half closed position. Because the

lag

time

is _actually longer than some of the exposure

times,

the

electronic sensor must anticipate the

lag

time and include it

when _making a decision about when to close.

Consequently,

variations in

lag

time from that designed into the

electronics will cause errors in exposure. This necessitates

an adjustment to the camera

during

manufacturing. The

lag

time is measured and, if

incorrect,

the shutter _spring, that

is the torsion _spring which holds the blades

together,

is

moved to a different anchor position. Then the

lag

time must

be remeasured to confirm performance. If the variations in

lag

time could be reduced, the _costly process of adjustment

and remeasuring would also be reduced. The optimization

objective of this _study is to minimize the variation in

lag

The shutter system to be considered consists of an

opening

blade,

a _closing

blade,

a shutter _spring, an armature

lever _assembly, and a magnetic

holding

coil. Figure 2.1a

shows the system in the initial covered aperture position.

The _opening blade covers the aperture and the shutter _spring

is _only

lightly

loaded. In Figure 2.1b the _opening blade has

been driven to its open position

by

the upward movement of an

opening lever which protrudes through a slot in the plane

supporting the shutter blades. Before the _opening lever

moves _upward, the magnetic

holding

coil is energized and

prevents the armature lever _assembly from _rotating

counterclockwise. This in turn prevents the _closing blade

from _rotating clockwise

by

_creating an interference between

the follower on the armature lever and the cam edge on the

closing blade. With the blades in this position, the shutter

spring is

fully

loaded,

and is _urging the _closing blade

clockwise. When the exposure is complete, the magnetic

holding

coil is deenergized and the _closing blade is allowed

to rotate clockwise to the position shown in Figure

2.1c,

covering the aperture. To complete the cycle, the opening

lever moves downward, rotating the two blades back to the

normal positon.

Armature

Magnetic

Holding

Coil

Openimg

LEVEf?

H

Armatuhe lever

OLUOWER

Cam

Edoe

Opening-"Blade

2.

Id

Covered

Aperture

Position

Before

Exposure:

Armature*

magnetic

Holding

Coil

s

Opening

Lever

Armature

Lever

Follower

Cam

Eqqe

2.1b

Opem

Aperature

P05ITION

Armature Lever

MAGNETIC

HOLDiMG

COil_\

Armatures

Open

ikg

Lever

Closinc-Blade

2.

1

c

Covered

Aperture

Position After

opening

progressive punch-and-die from 0.008 inch thick cold rolled,

hard

tempered,

type 301 stainless steel. The thickness and

the hardness of the material are _necessary to withstand

repeated impacts with the _opening lever and the _closing

blade. The _closing blade is also a punch-press _part, but it

is made of cold _rolled, half hard

tempered,

301 stainless

steel, and is 0.005 inches thick. The softer material is

required to produce a number of forms in the part with _sharp

bend radii. Both blades are required to complete their

travel in 4 to 5 milliseconds, so their moments of inertia

must be minimized. The _opening blade is moved

by

the _opening

lever,

which is propelled

by

a _{strong spring,} and which has a

relatively high momentum when it strikes the blade. The

closing blade is moved

by

a _relatively weak shutter _spring,

so its moment of inertia must be lower than that of the

opening blade to achieve the necessary travel times. Hence,

it is made of thinner material than the _opening blade.

The shutter spring is a one-turn torsion _spring made

from a 0.012 inch diameter hardened stainless steel wire.

The single turn is dictated

by

space limitations in the

camera, and the wire size is a result of design optimization

to produce the torques needed, the angular displacements of

the parts, and the lowest spring rate possible. The armature

glass fill for _rigidity and the PTFE for lubricity. Attached

to the armature lever is the armature. It consists of a

50 percent nickel and 50 percent iron _alloy in a powdered

form which has been pressed in a mold and sintered. This

material was chosen for its high magnetic permeability,

attaining a maximum

holding

force from the magnetic coil with

a minimum mass armature.

There are a number of parameters which will influence

lag

time if

they

are varied. Some of these parameters are

subject to control through design or through the

manufacturing process and some are not. Control cannot be

effectively exercised over the frictional forces.

Hence,

materials, processes, and coatings are used which will

minimize the coefficient of friction between _contacting

surfaces. Both of the shutter blades slide on a surface

consisting of glass-filled polystyrene. To reduce friction

at this interface the blades are painted with a low friction

paint. The follower on the armature lever _assembly slides on

the cam edge of the closing blade as the blade is

being

released. The PTFE fill in the plastic armature lever

reduces friction at this point. Friction at the pivot for

the shutter blades is minimized

by

molding the pivot from

acetal copolymer, a thermoplastic with high

lubricity

and

manufacturing process. The moment of inertia of the _closing

blade will _vary

directly

with the variation in the thickness

of the metal it is made from. The thickness of the paint on

the blade is assumed to be insignificant because its

density

is much less than that of the metal. Variations in the

densities of the metal and the paint are also assumed to be

insignificant. Because of the punch-press process

by

which

it is made, changes in moment of inertia due to changes in

the other two dimensions of the blade will be _very small and

are assumed insignificant.

The moment of inertia of the armature lever _assembly is

a combination of the properties of the armature lever and the

armature. The armature lever is injection molded from

polycarbonate, and therefore has excellent _{repeatability} in

its

dimensions,

as does the sintered armature. Variations in

the

density

of the polycarbonate are assumed small enough to

be ignored. The

density

of the sintered part is known to

vary considerably and it is located far from the pivot,

making it the only significant influence on the moment of

inertia of the armature lever assembly.

The torsional _spring constant and the _windup angle of

the shutter spring are also parameters that will _vary

significantly. Both of these will change due to variations

the _spring wire used. Variations will also result from the

set-up of the _{spring winding machines} and from the stress

relief treatment the springs receive.

A parameter which does not _{vary significantly}

during

the

manufacturing process but which is available for design

changes is the angle of the cam edge of the _closing blade

where it contacts the follower on the armature lever.

Because the part is made on a progressive _{punch-and-die,} the

angle of the cam edge is assumed not to vary. _However,

design changes made to this angle will have a significant

effect on

lag

time,

because it affects the relationship

between the angular acceleration of the armature lever and

the angular acceleration of the _closing blade.

In summary, variations in the

following

five design

related parameters are considered most significant for

affecting the variation of

lag

time for the _closing blade.

1. Thickness of the _closing blade.

2.

Density

of the sintered armature.

3. Torsional _spring constant of the shutter spring.

4.

Windup

angle of the shutter spring.

5. Angle of the cam edge on the _closing blade.

The effect of variations in these parameters will be

considered in what

follows,

as related to variation in

lag

CHAPTER 3

ANALYTICAL APPROACH TO DYNAMICS

3. 1 Calculation of Skin Friction on

Closing

Blade Due to Air

One item of interest in the _study of the _variability in

lag

time is whether there is a significant friction force

from the viscous

drag

of air on the _closing blade. In an

effort to gauge the order of magnitude of such forces in this

system, a worst-case approximate calculation is carried out.

The analysis will be made on the approximate _geometry of

figure 3.1.

Reference

1,

_pp

137,

gives equations and charts which

represent the

drag

forces on a rectangular plate as it is

moved edgewise through a viscous medium. The

drag

force is

given

by

equation 3.1.1.

D =

Cf

S

p

V*/2

(3.1.1)

In this equation, D is the total skin friction

drag

in

lbf,

C. is the

drag

coefficient, which is _unitless, S is the

wetted area of the plate in sq-ft,

p

is the

density

of the

fluid in slugs/cu-ft, and V is the _velocity of the plate in

ft/sec. The

drag

coefficient is given

by

the chart shown in

figure

3.2,

taken from reference

1,

_pp 137. It is shown as a

function of

Reynolds'

Number, which is given

by

equation

3.1.2.

Re =

p

V L

/

/a.

(3.1.2)

In this equation, Re is the

Reynolds'

Figure

3.

1

Approximations

used

for

Skin

Friction

Calculation

Figure

3.2

Drag-

_of

Smooth

Flat

Plate

-5kin

Friction

Reference

1,

pp

137.

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

\ 1

7TT ;i

i

! :

X

! \

|

'i.i, . . .',

\

1

:

i

_ -. ..

1

~ :

-";>-

&</

I

\v<

&K

j>

_DECREASING

^vr.

\f$\RM

$h

1

1 57"-^

EXTERNAL TURBULENCE

I

_Ill

104 105 106 107

REYNOLDS'

NUMBER=

unitless, L is the total length of the plate in the direction

of flow in

ft,

and ^i is the dynamic viscosity of the fluid

in slug/ft-sec.

To calculate the

drag

force,

some assumptions had to be

made to determine a worst-case approximation for velocity V.

Assuming

that there is information available about the time

required for the _closing blade to complete its _travel, from

elementary mechanics we have equation 3.1.3 for the average

angular _velocity of the _closing blade.

Wav = _A

/

t

(3.1.3)

In this _equation, Wav is the average angular _velocity in

degrees/sec,

A is the displacement angle in

degrees,

and t is

the elapsed time for that displacement in seconds.

Assuming

that the angular acceleration of the _closing blade is

constant, so that the angular _velocity changes

linearly,

and

that the _closing blade starts from _rest, we have equation

3.1.4 from _elementary mechanics.

Wmx = _{2 Wav}

(3.1.4)

In this equation, Wmx is the maximum angular velocity reached

in deg/sec. To convert from angular _velocity to linear

velocity we have equation 3.1.5 from _elementary mechanics.

Vmx = _{Wmx R}

(TT/180)

(3.1.5)

In this equation, Vmx is the maximum instantaneous linear

velocity in the tangential direction in ft/sec of a point

located a radius R ft from the center of rotation.

Combining

for the worst-case _velocity V to use in

determining

drag

forces.

V =

A R rr

/

90 t

(3.1.6)

Assuming

that the _closing blade travels 64 degrees from

a rest position in a time of 0.004 _seconds, we can calculate

a worst-case _velocity from the above equation. (The

analytical justification for this assumption can be found in

section

3.8)

The earlier assumption of constant angular

acceleration is conservative because the _closing blade is

driven

by

a torsion _spring with a

linearly decreasing

torque.

A constant angular acceleration would require a constant

torque,

so the actual maximum angular _velocity reached

by

the

closing blade will be lower than the approximation. The

radius to be used is that of a point

halfway

_along the

leading

edge of the large section of the shutter blade which

covers the aperture, as shown in figure 3.1. This radius is

chosen to represent the average speed of the large _section,

and its value is R = _{3.724 E-2} ft.

Substituting

the _above

values for A, t, and V into equation 3.1.6 we get _velocity V

= _{20.80 ft/sec.}

Reference

1,

_pp

65,

was used to find that for

dry

air at

59F and 14.7 psi the

density

f> is 2.378 E-3 slugs/cu-ft and

the dynamic viscosity ju is 3.73 E-7 slugs/ft-sec. An

approximate value for L was determined

by

using the largest

dimension of the closing blade in a direction approximately

:or

dimension occurs at

approximately

_the radius used above fc

the _velocity V.

Substituting

the values from above for

p

,

V,

L,

and ju

into equation 3.1.2 we obtain a Reynolds' Number of

Re = _5.526

E3.

Using

this number on the graph in figure 3.2

we see that it is below the range of values given for

Reynolds'

Number.

Looking

at the range of values given in

the graph for the

drag

coefficient

Cf

_, it is assumed that

the choice of a

drag

coefficient of .50 would be reasonably

conservative, since it is 50 times as large as the largest

drag

coefficient on the graph.

A value for

S,

the wetted area, is approximated

by

multiplying together the two largest dimensions of the

section of the _closing blade that covers the _aperture, and

multiplying

by

two for the two sides. The value obtained for

wetted area S = _{3.20 E-3 sq-ft.}

Using

_{the above values for}

C, , _p ,

V,

and S in equation 3.1.1 gives a value for

drag

force of D = _8.255 E-4 lbf.

Assuming

this force is applied

at the radius _R, from elementary mechanics the torque exerted

on the _closing blade would be given

by

equation 3.1.7.

T = _{D R}

(3.1.7)

In this equation T is the torque in ft-lbf.

Substituting

the

values for D and R into equation 3.1.7 gives a torque of

T = _{3.063 E-5} ft-lbf. The shutter _spring that drives the

closing blade has a maximum torque of Tmx

= _{2.465 E-3 ft-lbf.}

value of 1.2 percent. Since all the approximations that went

into the analysis were conservative of the

drag

forces,

and

since the

resulting

torque is so low compared to the other

torque

driving

the _closing

blade,

it is concluded that the

friction due to the viscous

drag

of air can be ignored in the

study of variations in

lag

time.

3. 2 Equation

Relating

the Angular Displacement of

Closing

Blade and Armature Lever

In order to write the dynamic equations for the _system,

the travel of the _closing blade in figure 2.1 is divided into

two parts.

During

the first part of travel the _closing blade

cam edge is _pushing on the follower from the armature lever

to rotate it counterclockwise. The shutter _spring is

accelerating both the closing blade and the armature lever

assembly

during

this period. The second part of travel

begins when the armature lever has disengaged from the

closing blade. The shutter spring now accelerates _only the

closing blade.

To analyze the first part of _travel, it is _necessary to

derive an equation which describes the angular displacement

of the armature lever as a function of the angular

displacement of the _closing blade. Since the angle of the

cam edge on the closing blade was identified earlier as a

useful parameter for adjusting the closing

time,

this angle

objective of this section is to derive an equation in the

form of equation 3.2.1.

<fi = _f

( T

,

Y)

(3.2.1)

In this _equation, <j> is the angular displacement of the

armature lever _assembly in the counterclockwise

direction,

F

is the angular displacement of the _closing blade in the

clockwise

direction,

and

Y

is the angle between the

horizontal and the cam edge of the _closing blade when it is

being

held back

by

the armature lever.

In figure

3.3a,

line 1 represents the cam edge of the

shutter blade when it is

being

held back

by

the armature

lever. Line 2 represents the cam edge of the shutter blade

as it rotates clockwise

by

an angle

^

. The point of

contact between the closing blade cam and the follower on the

armature lever is represented

by

the coordinates

(xo,

yo) . As

the blade _rotates, that point moves and becomes

(Xo,

Yo)

on

line 2. Changes in angle

Y

cause line 1 to rotate about

the point

(xo,

yo) and line 2 to rotate about the point

(Xo,

Yo)

. It is assumed that the actual changes in the point

of contact on the follower due to its circular shape are

small enough to be ignored. Coordinates

(xp,

_yp) represent

the location of the shutter blade pivot, and it is assumed

that yp = ₀. The angle

fi

represents the angle between the

vertical and the line connecting

(xp,

yp) with

(xo,

yo) , and

is given

by

equation 3.2.2 from elementary

geometry-ft

= arctan

|(xo

Figure

3.3

Geometry

of

Shutter System

3.3a.

Geometry

of

Closing

x

Blade

3.3b

Geometry

of

_>

Mmature

The radius from

(xp,

_yp) to

(xo,

_yo) is represented

by

r, and

is given

by

equation 3.2.3 from the Pythagorean theorem.

r =

y(xo

- xp)2

+ yo2

(3.2.3)

Coordinate Xo is given

by

_xp plus the product of r and the

sine of the sum of angles

ft

and <T _, given in equation

3.2.4.

Xo =

xp + r _sin(

fi

+ <T

)

(3.2.4)

Coordinate Yo is given

by

the product of r and the cosine of

the sum of angles j3 and

^

_, given in equation 3.2.5.

Yo =

r cos

(

fi

+

t

)

(3.2.5)

The slope M of line 2 is given

by

the negative of the tangent

of the sum of angles

Y

and <T in equation 3.2.6.

M = _-tan

( Y

₊

T

)

(3.2.6)

To find the equation for

B,

the y-axis intercept of

line

2,

the line equation is rearranged and the coordinates

(Xo. Yo) are subsituted

in,

to give equation 3.2.7.

B=Yo-MXo

(3.2.7)

Substituting

in the values of Yo, M, and Xo from equations

3.2.5,

3.2.6,

and 3.2.4 we get equation 3.2.8 for B as a

function of

T

and

Y

.

B = _{r cos(y+^)} + tan(r +

n(xp

+ r sin(/3 +

^))

(3.2.8)

where:

/$

= _arctan

((xo

-xp)

/

yo]

(3.2.2)

r =

/(xo

-xp)2

+ yo2

(3.2.3)

Using

the form _y = _{M x} + B and _substituting equations 3.2.7

and 3.2.8 for B and M gives equation 3.2.9 for line 2 as a

where:

y =

x _tan(-(f +

JT))

+ r cos(/S+<T) +

tan( f

+ JT

)

(xp

+ r sin(/5 + JP )

)

(3.2.9)

/* = _arctan

[(xo

-xp)

/

yo]

(3.2.2)

r =

/(xo

-xp)2

+ yo2

(3.2.3)

In order to reduce equation 3.2.9 to a more manageable

form,

dimensions from the shutter mechanism will now be

substituted into it. These dimensions are in millimeters,

but since the final equation will be _strictly a _relationship

between angles, the linear units will

drop

out. The

following

dimensions are to be substituted in.

= _15.62

mm yo = _{16.49 mm}

xp = _{10.7 mm}

yp = _{0.0 mm}

Substituting

these values into equation 3.2.9 gives equation

3.2.10 for line 2 as a function of

P

and

Y

.

y = _{x tan(-( V+}

J1)

_{+ 17.21 cos}

(

^

₊₁₆.

61)

+

tan( Y +

IT)

(10.7 + 17.21 sin (* +15.95) )

(3.2.10)

Since we now have an equation for _y as a function of

angles

T

and

Y

, we need an expression for <p as a

function of x and y. Figure 3.3b illustrates the geometric

relationship between the pivot and the follower on the

armature lever. The coordinate axes are located in the

same place relative to the shutter mechanism as

they

were

in figure

3.3a,

so the dimensions are compatible. The fact

that the contact point

(xf,

yf) of the follower is _very

nearly at the same _y dimension as the _pivot, coupled with the

disengages from the

closing

blade,

_{leads to some}

simplifications. The path of the contact point

(xf,

yf) of

the follower is _approximated

by

assuming that it follows a

straight _{vertical line for the 6 degree}

travel we are

interested

in.

The equation for this line is given

by

equation 3.2.11.

x = _xf

(3.2.11)

It is also assumed that the pivot has _exactly the same _y

dimension as the contact point. The displacement angle of

the armature lever then is given

by

equation 3.2.12.

jzi = _arctan

(y

-yf)

/

xf

(3.2.12)

To _simplify this expression, the

following

dimensions from

the shutter mechanism will be substituted.

xf= _{15.62 mm yf =16. 49 mm}

These values are the same as the previous xo and yo because

these points are in contact when the _closing blade is

being

held back

by

the armature lever.

Substituting

these values

into equation 3.2.12 gives equation 3.2.13 for the rotation

angle of the armature lever as a function of y.

0

= _arctan

(y

-16.49)

/

15.62

(3.2.13)

Equation 3.2.11 can now be substituted for x into equation

3.2.9. Then equation 3.2.9 can be substituted into equation 3.2.13 to give equation 3.2.14.

0

= _arctan

[(1/15.

62)

{l

5. 62 tan(-

(

P

+

Y )

)+17.21 cos(^

+16.61)

+

[tan

(6 +

Y)]

[id.

1+11.21 sin

(

t

+16.

6.1)]

-16.

49}

]

(3.2.14)

armature lever as a _function of the angular displacement of

the _closing blade and the cam angle of the _closing blade.

This equation will be used in later sections in the

derivation of the dynamic equations for the _closing blade.

3. 3 Determination of Disengagement Point

In the dynamic equations for the closing blade travel,

it is _necessary to know the angle of closing blade travel at

which the armature lever disengages from the _closing blade.

The effective moment of inertia of the _closing blade and

armature lever together is much greater than that of the

closing blade alone, so there will be a significant change in

the angular acceleration at this point. The angle of

rotation of the armature lever 0d at which it disengages

will remain fixed because it is limited

by

the _geometry of

other parts. As the cam angle

Y

is changed, the angle of

rotation of the _closing blade required to reach the

disengagement point will change. As the cam angle

increases,

the armature lever follower will be pushed upward more

rapidly, so the disengagement angle

^d

will decrease. The

converse is also true.

In order to determine the disengagement angle for _any

value of cam angle

Y

, equation 3.2.14 must be solved for

the closing blade travel

<5.

Upon inspection of the

equation it was concluded that _putting it in an explicit form

Therefore,

a

root-finding

_{method was used in all subsequent}

calculations.

A program was _{written for an HP-41CV calculator and the}

disengagement

angle

)fd

was determined for 0d = _{6 degrees}

and for several different values of cam angle

Y

. A flow

chart of the program can be found in Appendix A. The same

cases were done

by laying

out the parts on a

drawing

board

and

finding

a graphical _{solution. The} comparison of the

calculated values with the graphical values for disengagement

angle <^d are shown in figure 3.4. There is a consistent

disagreement between the two methods of about 0.5

degrees,

which is assumed to be due to the inherent

inaccuracy

of a

layout and the bias of the

drafting

machine used.

3.4 Approximation for

Coupling

of Armature Lever and

Closing

Blade

During

the first part of shutter blade

travel,

the

angular displacements of the _closing blade and armature lever

assembly are coupled together. For the purposes of later

calculations, it is desirable that there be a direct ratio

between the angular displacements of these two parts. _If,

for example, the angular displacement of the armature lever

is always one half that of the _closing

blade,

then the

angular _velocity of the armature lever will always be one

half that of the closing blade.

Similarly,

the angular

Figure 3.

M-

Comparison

of

Calculated

vs.

Graphical

VALUE5

OF

Di5ENGA6EMENIT

AN6LE

OF

Closing

Blade

Yd

UJ oc

UJ

So

<

a.

<

or

5 6 7 8 9 10 _n 12. 13 1* IS It a 18

that of the

closing

blade.

_This

only holds true if the

relationship between

the angular displacements of the two

parts is

linear.

A good example is that of two gears of

different pitch

diameter

meshed _together. Their angular

displacements,

angular _velocities, and angular accelerations

are all _linked

by

_{the same}

ratio.

Figure 3.5 shows the angular displacement of the _closing

blade versus that of the armature lever _assembly for three

values of cam angle Y . The cam angles _of 24 degrees _and

46 degrees represent the lower and upper extremes possible

based on the _geometry of the system. The 36 degree cam angle

is _{approximately} the middle value. It can be seen that the

relationship between the angular displacements of the _closing

blade and the armature lever calculated

by

equation 3.2.14

are not linear. The straight lines on each graph represent a

linear approximation to the _relationship between the angular

displacements. The ratio used for the straight lines is the

ratio between the total angular displacements of the two

parts at the disengagement point.

In order to evaluate the error that would be introduced

by

using the linear approximation, the error in armature

lever angular displacement <j) was calculated at each of ten

equally spaced locations along the curves. These errors were

then averaged for each case, and the averages were divided

by

the mean value for </> , which is 3 degrees. The relative

Figure

3.5

6

5

-^*

1X1

LU ,

a:

3

o

Q 2

^-^

^

in"

2

z

o

^

5

?

ui 3

or

^Sl

/ANGULAR

DlSPLflCENENT

OF

ARMATURE

LeVC*

^

VS.

Angular

Displacement

of

Closing

Blade

9

For

Various Cam Angles V

y--2f

degrees

A

=0.637

DEGREES

8 q 10 II 12 13 <<f 13 lb

0

(degrees)

36

DEGREES

A

=0.429

D&&R&E5

LINEAR

EQUATION

3.2.IY

J 3*56781

9

(DEGREES)

^

s₄₆

DEGREES

&=

CX42f DE6KE

j i L

to il IZ 13 if 15 16

LINEAR

EQUATION,

3.2.IM-j i 1 u J I I L J L

I t 3 io n it 13

if-15 lb

21.2 percent for cam angle

Y

=24

degrees,

14.3 percent for

Y

= ₃₆

degrees,

and 14.1 percent for

Y

= _{46 degrees.}

The

following

_{observations are}

made _regarding the use of the

linear _{approximation to the}

angular displacement.

1. The errors in angular

displacement,

and therefore

angular _{acceleration, occur over a small percentage of}

the total travel.

2. The linear approximation used agrees _exactly with the

actual angular displacement at the disengagement

point, which is a critical point for later

calculations.

3. The objective of the _study is to model the variation

in

lag

time,

rather than the absolute

lag

time.

These facts constitute a reasonable

justification

for the

use of the linear approximation in later calculations.

3.5 Equation of Motion for First Part of Travel

The first part of travel of the _closing blade is defined

as the travel from the position where it is

being

held back

by

the armature lever to the position where it disengages

from the armature lever.

During

this travel the shutter

spring is accelerating both the closing blade and the

armature lever assembly. Since friction forces are not known

and are _relatively small,

they

will be assumed to be

negligible.

for a _{mass, spring, and damper}

system with linear travel and

no

driving

function

_{is given}

by

equation 3.5.1.

m d*_x +bdx+kx=0

(3.5.1)

dt* _dt

In this _equation, m is the mass, x is the

displacement,

t is

the

time,

b is the

damping

_coefficient, and k is the _spring

constant. Each term represents a

force,

and the sum of the

forces equals zero.

To use this equation for shutter blade _travel, the term

for

damping

forces is assumed to be _{relatively small,} and the

equation is rewritten in terms of _rotary motion, giving

equation 3.5.2.

I

d^0_

+ kO = ₀

(3.5.2)

dt*

In this equation, I is the moment of

inertia,

6

is the

angular

displacement,

t is the _time, and k is the torsional

spring constant. Each term represents a _torque, and the sum

of the torques equals zero.

To solve this differential equation we first rearrange

it,

giving equation 3.5.3.

d*0 + k6 = ₀

(3.5.3)

dt2 _I

Letting

= k/I _{we get} equation 3.5.4.

d26 +u26 = _{0 (3.5.4)}

dt*

The characteristic equation for this is given

by

equation

r2

+ W*

=0

(3.5.5)

In this equation r is a

dummy

variable.

By

inspection the

i -

Z"1*

The solution to equation 3.5.4 is therefore given

by

equation

3.5.6.

0(t)

=

c,

exp(iwt) + _c2 exp(-i^t)

(3.5.6)

In order to convert this expression from complex form to real

form we use Euler's relations from reference

2,

_pp 64.

exp(ibx) =

cos bx + i sin bx (3.5.7)

exp(-ibx) = _{cos bx} - i

sin bx

(3.5.8)

Applying

equations 3.5.7 and 3.5.8 to equation 3.5.5 gives

equation 3.5.9.

0(t)=ct

(coswt + i sinwt) + c2(coswt - i

sinut)

(3.5.9)

Rearranging

this equation we get equation 3.5.10.

0(t)

=

(c,+c2)

_{cos uj t} + i (c,-c2) sinwt

(3.5.10)

If we let

C,

=

c, + _c2 and

C2

= _i

(c,

-c2 ) , we can write

the solution to equation 3.5.4 in the form of equation

3.5.11.

@(t)

=

C,

coswt + C2sina;t

(3.5.11)

The initial conditions on this problem are that the

angular _velocity equals zero and the angular displacement is

known,

and is designated

0(0).

If we write equation 3.5.11

at time t = ₀

, we get equation 3.5.12.

0(0)

=

C,

cosw0 +

C2

sinw0 (3.5.12^

Solving

this for

C,

we get C =

0(0)

.

Differentiating equation 3.5.11 and _writing it for time t = ₀

gives equation 3.5.13.

0 = -

0(0)

WsinW0 +

Cz

wcosy0

(3.5.13)

The angular

displacement

_{as a function}

of time for the

closing blade is given

by

the complete _solution, equation

3.5.14.

0(t)

=

0(0)

_coso/t

(3.5.14)

Turning

our attention to w _, we recall from equation

3.5.4 that we set u)1

= _k/I

. The moment of inertia in effect

during

the first part of travel is the moment of inertia of

the _closing blade combined with that of the armature lever

assembly. It was stated in section 3.4 that the angular

acceleration of the _closing blade and the angular

acceleration of the armature lever _assembly would be assumed

to be related

by

a ratio. The first term in equation 3.5.2

is the product of the moment if inertia and the angular

acceleration. Since that term is

linear,

the ratio between

the angular accelerations can be applied to the moment of

inertia of the armature lever _assembly and achieve the same

result.

For example, gear A and gear B are meshed together and

gear A has a larger pitch diameter and a larger moment of

inertia than gear B. For the purpose of _writing the dynamic

equation, gear B _may be treated as if it has the same pitch

diameter,

and therefore the same angular acceleration, as

gear A if the moment of inertia of gear B is multiplied

by

the ratio of the pitch diameters, pdB/pdA. The equation for

undamped motion for the two gears can be written as shown in

[iA

+ IB

(pdB/pdA)]

d2

(

6

A)

+

k(

0A)

= ₀

(3.5.15)

Jt*

In this _equation, IA is the moment of inertia of gear

A,

IB

is the moment of

inertia

of gear

B,

pdA is the pitch diameter

of gear

A,

pdB is the pitch diameter of gear _B, and

0A

is

the angular _{displacement of gear A. The term in square}

brackets in equation 3.5.15 will be called the "equivalent

moment of

interia",

Ieq.

In a similar _manner, an equation _may be written for an

equivalent moment of inertia for the _closing blade and

armature lever assembly. The moments of inertia of the

closing blade and the armature lever assembly are equated to

the moments of inertia of gears A and B. The ratio of

angular displacements of the closing blade and the armature

lever

(

^

d/

^d)

is equated to the ratio of pitch diameters of

gears A and B

(pdB/pdA)

, giving equation 3.5.16 for the

equivalent moment of inertia.

Ieq

= _{Icb + lal(*d//d)}

(3.5.16)

In this equation, Icb is the moment of inertia of the _closing

blade,

Ial is the moment of inertia of the armature lever

assembly, <f> d is the angular displacement of the armature

lever assembly at disengagement, and

^d

is the angular

displacement of the closing blade at disengagement.

In equation 3.5.14 we set

uj2

= _{k/I. It follows that}

t is given

by

equation 3.5.17.

oj t =

/k

_ta/I

(3.5.17)

gives equation 3.5.1R for the angular displacement of the

closing blade as a function of time and as a function of the

parameters to be varied.

6(t)

=

(0)

cos^k t*/(Icb + Ial

( 0

d/

3M)

)

(3.5.1^)

In ordar to determine the travel time for the second

part of _closing blade

travel,

it will be _necessary to know

the angular _velocity of the _closing blade at disengagement.

Equation 3.5.19 for the angular _velocity of the _closing blade

is obtained

by differentiating

equation 3.5.18 with respect

to time. (3.5. 19

below)

J7(t)=-(9(0)/k/(Icb+Ial( 0

d/

^d)

) sin/kt2/(Icb+Ial(

$

d/

**d)

)

In this _equation, _/7(t) is the angular velocity of the

closing blade

during

the first part of travel.

3. 6 Equation of Motion for Second Part of Travel

The second part of travel of the _closing blade is

defined as the angular displacement from the disengagement

point to the position where the _closing blade is _covering

half of the aperture. The applicable differential equation

is again the rotary version of the mass, spring, and damper

equation with the

damping

term removed, shown below as

equation 3.5.1.

I d20 + k0 =0

(3.5.1)

dt2

The general solution to this equation, which was developed in

section

3.5,

is given in equation 3.5.2.

This problem has

initial

conditions of a known displacement

0(d)

and a known angular _velocity

fl

(d)

_at time t = _{0, which}

is at the

disengagement

_point.

Writing

equation 3.5.2 for t

= _{0 gives}

equation 3.5.3.

0(d)

=

C,

coswfl +

C2

sincj0

(3.6.3)

Solving

this for

C,

gives

C,

=

0(d)

.

Differentiating

equation 3.6.2 to get the angular _velocity

equation and _writing that equation for time t = ₀

gives

equation 3.5.4.

fl

(d) =

-0(d)

wsinyd +

Cz

wcoso/0

(3.6./1)

Solving

this for

C2

gives

C2

=

fl

(d) /co

.

Substituting

the

two constants back into equation 3.5.2 gives the final

solution, equation 3.6.5.

0(t)

=

0(d)

_{coswt + (}

fl

(d)

sincJtWuJ

(3.5.5)

From section 3.5 we recall that w = _{k/I. In this}

case

I is

Icb,

the moment of inertia of the _closing blade.

Writing

equation 3.5.5 in terms of parameters that can be

varied to change _closing time we get equation ^.6.6.

0(t)

=

0(d)

_cos/kt2/Icb + _(l(d)/lcb/ksin/kt2/Icb

(3.6.6)

This equation, together with equation 3.5.18 can be used to

determine the total time required for the _closing blade to

move from the latched position to the half-closed position.

3. 7 Calculation of Moments of Inertia

The purpose of this section is to illustrate the

blade and the armature _{lever assembly. No recent}

measurements of these moments were _available, so

they

had to

be calculated.

The shape of the _closing blade does not lend itself to

being

broken _up into simple geometric shapes with polar

moments of inertia that can be _readily calculated and then

combined to determine the total moment of _inertia. A method

was therefore devised for _calculating polar moment of inertia

for _any planar _irregular shape. A calculator program is

shown in appendix B which is written for the HP-41CV and

which calculates polar moment of inertia for a planar part

by

the

following

method.

An accurate layout of the part is required with as large

a scale as practical. A line is drawn which passes through

the point about which the polar moment of inertia is

required. This is shown as "first line"

on figure 3.6. For

convenience the line should pass _{approximately} through the

center of the shape to be considered. Lines are then drawn

at regular intervals perpendicular to the first line and

extending past the edges of the part in both directions.

Judgement must be used in

deciding

the size of the interval

between the

lines,

with a smaller interval _providing greater

accuracy. Three measurements are then made for each cross

line,

the distance _along the first line from the point of

interest to the cross line, and the distance _along the cross

First

Line

Cross Lines

passes the edges of the part to the right and to the

left,

as

shown in figure 3.6. The calculator program prompts the user

for the _{interval between cross}

lines, and then prompts

repeatedly for the three measurements for each cross

line,

displaying

a progressive sum of the polar moment of inertia

for each cycle. The program calculates the moment of inertia

of each narrow strip, shown in the figure

by

a dashed _line,

about the first line and about a second line perpendicular to

the first line and _passing through the point of interest.

The polar moment of inertia of each _strip is determined

by

adding these two moments together. The polar moments of the

strips are then _{progressively} summed. The method is

straightforward, and the _accuracy can be adjusted

by

the user

as he deems necessary.

The polar moment of inertia of the irregular part of the

closing blade was determined using the calculator program and

this was added to the polar moment of the annular ring

obtained through standard formulas. The result of 3.2R5 E-6

oz-in-sec2

compared

favorably

with physical, measurements of

polar moment of inertia made on obsolete versions of this

pa rt.

The polar moment of inertia of the armature lever was

calculated

by

_treating it as two rectangular solids, each

with the same length and approximately the same

cross-sectional area as one of the arms of the part. The

pivot was calculated and _{found to account for} 57 percent of

the moment of _{inertia of the assembly- The moment of inertia}

calculated was 1.151 E-5 oz-in-sec , which is 3.5 times as

large as that of the _closing

blade,

_showing that the armature

lever will have a significant effect on the first part of

travel of the _{closing blade.}

3. 8 Example Calculation of

Closing

Time

The objective of this section is to go through the

process of _{calculating closing} time with parameter values

from a known case and to compare the results with known

values. In the last section the moments of inertia of the

closing blade and the armature lever _assembly were

calculated. The other values needed for the calculations are

the torsional _spring constant

k,

which is 0.314 oz-in/radian,

the _windup angle of the _spring

0(0),

which is -1.507

radians, and the cam angle of the _closing blade Y , which is

0.663 radians. These values come from measurements of

manufactured parts.

First the disengagement angle of the _closing blade must

be determined _using equation 3.2.14 which is shown below as

equation 3.8.1.

0

= _arctan

[(1/15.62)

{l

5. 6? tan

(

-(

V + Y

) )

+1"?. 21 cos

(

V +16.

61)

+

[tan(f+r')]

[l0

. 7+17. 21 sin (

Y

+16.

51)]

-16.

49}]

(3.8.1)

In this equation,

#

is the angular displacement of the

armature lever assembly,

X

is the angular displacement of the

It can be seen

by

_examination

of this equation that it would

be

extremely

difficult

to write it _explicitly for

the angular

displacement

_{of the}

closing blade

Y

. It is

therefore

n-cessary

_{to use a}

root-finding

technique to solve

for

Y

as a

function

_of

0

,-md

Y

. The technique _chosen

was

bisection,

_{as described in}

reference

3,

_pp

21,

because it

will always converge to a solution. A calculator program for

the HP-41CV was written to do

this,

and is shown in appendix

A.

The angular _displacement of the armature lever at the

disengagement point jzf d , is 6 degrees (.105 _radians) and is

limited to this value

by

the _geometry of other parts.

Using

this program, the angular displacement of the _closing blade

at the disengagement point

Td

was determined to be .153

radian

The next equation needed is equation 3.5.18 for the

first part of shutter blade _travel, which is rewritten

explicitly for time t as equation 3.8.2.

t =

/(1/k)

[icb

₊

IaM^d/fd)]

[arccos

(0(d)

/0(H)]

(3.8.2)

In this equation, t is in seconds, k is the torsional _spring

constant of the shutter _spring in oz-in/rad, Icb is the

moment of inertia of the _closing blade in oz-in-sec , Ial is

the moment of inertia of the armature lever _assembly in

oz-in-sec , and

0(0)

is the angular displacement of the

shutter _spring from no load to the latched position. The

the disengagement _point,

0(d)

_{-an be} found

by

_adding

Y

d _to

0(0).

n4nce

0(T)

is a negative angle and

^d

is a positive

angle,

0(d)

will be a smaller negative angle.

Substituting

in the parameters gives 2. ^2 E-3 seconds as the time for tha

first part of _closing blade travel.

To get the time for the second part of shutter blade

travel it is first _necessary to use eguation 3.5.19,

rewritten as equation

3.8.3,

to determine the angular

velocity of the _closing blade at the disengagement point.

flit)

= -

0(0)/k/(Icb+Ial

(

0

d/

/d)

)

sin /kt2 /(Icb+lai ₍

₀

d/ <T d) ) (3.8.3)

Substituting

into this equation the parameters determined

above gives an angular _velocity

il(d)

at the disengagement

point of 114.18 rad/sec.

The next equation needed is equation 3.6. s for the

second part of _closing blade travel as a function of time,

rewritten as equation 3.8.4.

0(h)-

0(d)

cos/kt2/Icb + J?(d)/lcb/k sin/kt2/Icb

(3.3.4)

For substitution into this equation

0(h)

is the angular

displacement of the shutter _spring from no load to the half

closed point, and has a value of -1.083 radians.

Upon inspection of this equation, no _way was found to

rearrange it to give "t" explicitly. _Therefore, a

root-finding method was used to find a solution. The

function is a sinusoid, and the root of interest is the

risk that the

initial

interval

_{chosen will contain more} than

one root and that the method will not converge to the

smallest root. A method was therefore

devised,

for the

optimization problem to

follow,

to choose the initial

interval. This is done

by

_{stepping along} the positive axis

from zero until a change is detected in the sign of the

function,

indicating

that a root had been passed. Tha1: _step

is then taken as the right

boundary

for the initial interval

and two steps are taken backwards to establish the left

boundary. The bisection method is then applied _normally to

this interval .

Using

this method on equation 3.3.4 for the parameters

of this example gave a time for the second part of _closing

blade travel of 1.35 E-3 seconds.

Adding

this to the time

calculated for the first part of travel of 2.72 E-3 seconds

gives a total _closing time of 4.07 E-3 seconds. The time

measured for mechanisms with these parameters is 4.3 E-3

seconds, giving an error of 5 percent. This small error

indicates that friction does not have much effect on the

closing time and that the other assumptions and

approximations that were used in the derivation of the

CHAPTER 4

APPROACHES TO OPTIMIZATION

4. 1 The Optimization Problem

The most common perception of an optimization problem is

one in which some specific measureable performance parameter

is to be maximized or minimized. For this problem one might

initially

expect that the time required for _closing blade

travel should be _minimized, or that the force required from

the _opening lever or the magnetic

holding

coil should be

minimized. Even though a fast shutter time is desirable for

accurate _exposures, in this case there are synchronizations

to be considered with other

functions,

such as the

firing

of

the electronic

flash,

which are dependent on a specific

closing time for proper performance.

If the system as it exists

today

is modified _slightly

in one of its parameters it is possible to compensate

by

changing another parameter and achieve the same _closing

time-For example, if the moment of inertia of the closing blade is

increased

by

making it from a thicker metal the closing times

would increase. However, the windup angle of the shutter

spring might be increased to provide more torque to drive

the

blade,

bringing

the _closing time back to its original

value- Since _any pair of the five relevant parameters could

be adjusted in this manner it can be seen that there are an

would provide the proper

closing

_time.

Each of the five parameters described in Chapter 2 is

subject to tolerances in the _{manufacturing} process. The cam

angle on the _closing blade is subject to _very little change,

because it is made on a _{progressive punch and die. The}